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Consider the following statement which is a part of Conjecture 1.3 in the paper titled "The asymptotic growth of torsion homology for arithmetic groups" authored by N. Bergeron and A. Venkatesh.

Let $G$ be a $\mathbb{Q}$-semisimple group of zero defect(includes the Shimura variety case) and let $K \subset G(\mathbb{R})$ be a maximal compact subgroup and $\Gamma \subset G(\mathbb{Q})$ be a congruence subgroup. Denote the locally symmetric space $\Gamma \backslash G(\mathbb{R})/K$ by $X_{\Gamma}$. Let $\Gamma_N$ denote the descending family of congruence subgroups such that $\cap \Gamma_N = \lbrace 1 \rbrace$. Then $$ \lim_{N \to \infty} \frac{\log |\mathrm{H}^i(X_{\Gamma_N}, \mathbb{Z})_{tors}|}{[\Gamma : \Gamma_N]} = 0. $$

I was wondering what are heuristics for this conjecture? What are the evidence available for this conjecture? Is something known in the case of Shimura varieties? Any help of reference is appreciated. Thank you.

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    $\begingroup$ Can one replace $\log$ be a root of high enough degree (independent of $i$)? $\endgroup$
    – markvs
    Aug 9, 2021 at 19:46
  • $\begingroup$ @MarkSapir Thank you for your comment. I have no idea what the answer should be but that is an interesting question since a modification of the sort you mention might make the conjecture even stronger than the one mentioned above. Also makes me wonder can one replace the bottom term in LHS by a root of high enough degree(independent of i and N?) and get a non-zero number in the RHS? This one also seems stronger than the one written in the question above. $\endgroup$
    – random123
    Aug 10, 2021 at 7:11

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